I keep seeing conflicting definitions of a statistic. Is a statistic a random variable such that it is a function of the random variables of a random sample? Or is it the value of the function of the random sample, after each random variable of the random sample has taken on a specific value?

$$(1) \ S=f(X_1,X_2...X_n)$$

$$(2) \ s=f(x_1,x_2...x_n)$$

I haven't been able to get any clarification for this and I've seen the term statistic describe both situations

  • $\begingroup$ It seems that there are also plenty of conflicting views on estimator and estimate which go hand in hand with this. As an estimator is supposed to be a kind of statistic, whether an estimator is a random variable or not would also clarify some things. There seems to be a lot of conflicting definitions on this site though. $\endgroup$ Commented Mar 11, 2019 at 21:22

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A statistic is a function that maps from the set of outcomes of the observable values to a real number. Thus, with $n$ data points, a statistic will be a function $s: \mathbb{R}^n\rightarrow \mathbb{R}$ as in your second form. However, it is also possible to view the statistic in its random sense by taking the appropriate composition of function with the original random variables. (Remember that each random variable $X_i: \Omega \rightarrow \mathbb{R}$ is a measurable function that maps from the sample space to the real numbers.) That is, you can form the random variable $S: \Omega \rightarrow \mathbb{R}$ as:

$$S(\omega) = s(X_1(\omega), ..., X_n(\omega)).$$

The random variable $S$ is the random version of the statistic $s$. Both are often referred to as "statistics", but it is important to bear in mind that $S$ is a composition with the functions for the observable random variables.

  • $\begingroup$ that was very helpful. A lot of this notation is really confusing and seems at time almost conflicting as in this case where the term can be used in both contexts. $\endgroup$ Commented Mar 11, 2019 at 21:38

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